Formal geometry proofs

1. Proofs of triangle and polygon theorems


ID is: 1857 Seed is: 1074

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as M=90°, N=x, and P=45°. Sides MN¯ and MP¯ are equal in length, as indicated. An exterior angle y is formed by extending one side of the triangle toward point Q.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
2 attempts remaining
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°90°+x+45°=180°
x=180°90°45°x=45°
NOTE: You can also solve for x using the fact that this triangle is isosceles (it has two equal sides). Isosceles triangles have two congruent (equal) angles. In this case the equal angles are N and P. And that means that x is equal to 45°.

Angle x has a measure of 45°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle QM^P is equal to the sum of the opposite interior angles, N and P. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=N+Py=45°+45°y=90°
NOTE: You can also find y using supplementary angles. The two angles at M must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=90°.

The correct answers are:

  1. x= 45°
  2. y= 90°

Submit your answer as: and

ID is: 1857 Seed is: 9480

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as M=83.62°, N=48.19°, and P=x. Sides MN¯ and MP¯ are equal in length, as indicated. An exterior angle y is formed by extending one side of the triangle toward point Q.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
2 attempts remaining
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°83.62°+48.19°+x=180°
x=180°83.62°48.19°x=48.19°
NOTE: You can also solve for x using the fact that this triangle is isosceles (it has two equal sides). Isosceles triangles have two congruent (equal) angles. In this case the equal angles are N and P. And that means that x is equal to 48.19°.

Angle x has a measure of 48.19°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle QM^P is equal to the sum of the opposite interior angles, N and P. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=N+Py=48.19°+48.19°y=96.38°
NOTE: You can also find y using supplementary angles. The two angles at M must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=96.38°.

The correct answers are:

  1. x= 48.19°
  2. y= 96.38°

Submit your answer as: and

ID is: 1857 Seed is: 6710

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as M=90°, N=45°, and P=x. Sides MN¯ and MP¯ are equal in length, as indicated. An exterior angle y is formed by extending one side of the triangle toward point Q.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
2 attempts remaining
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°90°+45°+x=180°
x=180°90°45°x=45°
NOTE: You can also solve for x using the fact that this triangle is isosceles (it has two equal sides). Isosceles triangles have two congruent (equal) angles. In this case the equal angles are N and P. And that means that x is equal to 45°.

Angle x has a measure of 45°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle QM^P is equal to the sum of the opposite interior angles, N and P. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=N+Py=45°+45°y=90°
NOTE: You can also find y using supplementary angles. The two angles at M must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=90°.

The correct answers are:

  1. x= 45°
  2. y= 90°

Submit your answer as: and

ID is: 1907 Seed is: 9473

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
2 attempts remaining
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 8 sides. This polygon is called a octagon.

The sum of the interior angles of this octagon = 1080 °.

Let n be the number of sidesn=8Sum of Int angles of a polygon=180(n2)=180×((8)2)=180×(6)Sum of Int angles of a polygon=1080°
Each individual angle=1080÷8=135°

Submit your answer as: and

ID is: 1907 Seed is: 5984

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
2 attempts remaining
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 6 sides. This polygon is called a hexagon.

The sum of the interior angles of this hexagon = 720 °.

Let n be the number of sidesn=6Sum of Int angles of a polygon=180(n2)=180×((6)2)=180×(4)Sum of Int angles of a polygon=720°
Each individual angle=720÷6=120°

Submit your answer as: and

ID is: 1907 Seed is: 9540

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
2 attempts remaining
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 7 sides. This polygon is called a heptagon.

The sum of the interior angles of this heptagon = 900 °.

Let n be the number of sidesn=7Sum of Int angles of a polygon=180(n2)=180×((7)2)=180×(5)Sum of Int angles of a polygon=900°
Each individual angle=900÷7=128.57°

Submit your answer as: and

ID is: 1905 Seed is: 8801

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
2 attempts remaining
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 5 sides. This is called a pentagon.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°5=72°

Submit your answer as:

ID is: 1905 Seed is: 29

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
2 attempts remaining
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 8 sides. This is called a ocatagon.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°8=45°

Submit your answer as:

ID is: 1905 Seed is: 7391

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
2 attempts remaining
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 8 sides. This is called a ocatagon.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°8=45°

Submit your answer as:

ID is: 1908 Seed is: 414

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.01))
2 attempts remaining
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 8 sides. This polygon is called a octagon.

Let n be the number of sides. Then n=8.

x=Size of Ext angle of polygon=360°n=360°8=45°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((8)2)=180°(6)=1,080°

We can use this to work out what each individual interior angle is.

interior angle=1,080°÷8=135°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°135°=45°

So x=45°


Submit your answer as:

ID is: 1908 Seed is: 4190

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.01))
2 attempts remaining
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 10 sides. This polygon is called a decagon.

Let n be the number of sides. Then n=10.

x=Size of Ext angle of polygon=360°n=360°10=36°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((10)2)=180°(8)=1,440°

We can use this to work out what each individual interior angle is.

interior angle=1,440°÷10=144°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°144°=36°

So x=36°


Submit your answer as:

ID is: 1908 Seed is: 6925

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.01))
2 attempts remaining
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 9 sides. This polygon is called a nonagon.

Let n be the number of sides. Then n=9.

x=Size of Ext angle of polygon=360°n=360°9=40°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((9)2)=180°(7)=1,260°

We can use this to work out what each individual interior angle is.

interior angle=1,260°÷9=140°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°140°=40°

So x=40°


Submit your answer as:

2. Congruency in triangles


ID is: 4194 Seed is: 6890

Prove congruency with a common side

In the diagram below, CBDA and CB=AD=8 cm.

Prove that ΔCBDΔADB.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔCBD and ΔADB:

  1. CB=AD=8 cm (given)

ΔCBDΔADB

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that CB=AD=8 cm, so we have a pair of equal sides.

We can see that the line BD is a side of both triangles. When a matching side is shared by two triangles, we say that the side is common. So, we have another pair of equal sides.

We were also told that CBDA, so we can use alternate angles between parallel lines to prove that CB^D=AD^B. So we have a pair of equal angles.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔCBD and ΔADB:

  1. CB=AD=8 cm (given)
  2. BD is common
  3. CB^D=AD^B (alt s; CBDA)

ΔCBDΔADB (SAS).


Submit your answer as: andandand

ID is: 4194 Seed is: 2177

Prove congruency with a common side

In the diagram below, BACD and BA=DC=9 cm.

Prove that ΔBACΔDCA.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔBAC and ΔDCA:

  1. BA=DC=9 cm (given)

ΔBACΔDCA

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that BA=DC=9 cm, so we have a pair of equal sides.

We can see that the line AC is a side of both triangles. When a matching side is shared by two triangles, we say that the side is common. So, we have another pair of equal sides.

We were also told that BACD, so we can use alternate angles between parallel lines to prove that BA^C=DC^A. So we have a pair of equal angles.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔBAC and ΔDCA:

  1. BA=DC=9 cm (given)
  2. AC is common
  3. BA^C=DC^A (alt s; BACD)

ΔBACΔDCA (SAS).


Submit your answer as: andandand

ID is: 4194 Seed is: 1898

Prove congruency with a common side

In the diagram below, BC=BD and BACD.

Prove that ΔBCAΔBDA.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔBCA and ΔBDA:

  1. BC=BD (given)

ΔBCAΔBDA

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that BACD, so we know that BA^C=BA^D=90°. So we have a pair of 90° angles.

We were also given that BC=BD, so we have a pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

We can see that the line BA is a side of both triangles. When the same side is shared by the two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔBCA and ΔBDA:

  1. BC=BD (given)
  2. BA is common
  3. BA^C=BA^D=90° (given)

ΔBCAΔBDA (90°HS).

NOTE: You might have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the two known sides.

Submit your answer as: andandand

ID is: 1858 Seed is: 9047

Congruent triangles

In the diagram below, ΔUVWΔUXW. Also, UWXV while VW=15 and UW=12.

  1. Calculate the value of x.
  2. Determine the length of XV.
Answer:
  1. x= units
  2. XV= units
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side UX.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that UWXV, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle UXW, but you can use either one, because they are congruent.

In ΔUXW:XW=15(ΔUVWΔUXW)(15)2=x2+(12)2(Pythagoras)225=x2+144225144=x2±81=x9=x

The length of UX is 9 units.


STEP: Find the length of XV
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment XV. This segment is twice as long as x, which we calculated above. So XV=2(9)=18 units.

The correct answers are:

  1. The length of side x is 9 units.
  2. The length of XV is 18 units.

Submit your answer as: and

ID is: 1858 Seed is: 582

Congruent triangles

In the diagram below, ΔABCΔADC. Also, ACDB while BC=30 and AC=24.

  1. Calculate the value of x.
  2. Determine the length of DB.
Answer:
  1. x= units
  2. DB= units
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side AD.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that ACDB, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle ADC, but you can use either one, because they are congruent.

In ΔADC:DC=30(ΔABCΔADC)(30)2=x2+(24)2(Pythagoras)900=x2+576900576=x2±324=x18=x

The length of AD is 18 units.


STEP: Find the length of DB
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment DB. This segment is twice as long as x, which we calculated above. So DB=2(18)=36 units.

The correct answers are:

  1. The length of side x is 18 units.
  2. The length of DB is 36 units.

Submit your answer as: and

ID is: 1858 Seed is: 4071

Congruent triangles

In the diagram below, ΔMNPΔMQP. Also, MPQN while NP=20 and MP=16.

  1. Calculate the value of x.
  2. Determine the length of QN.
Answer:
  1. x= units
  2. QN= units
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side MQ.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that MPQN, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle MQP, but you can use either one, because they are congruent.

In ΔMQP:QP=20(ΔMNPΔMQP)(20)2=x2+(16)2(Pythagoras)400=x2+256400256=x2±144=x12=x

The length of MQ is 12 units.


STEP: Find the length of QN
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment QN. This segment is twice as long as x, which we calculated above. So QN=2(12)=24 units.

The correct answers are:

  1. The length of side x is 12 units.
  2. The length of QN is 24 units.

Submit your answer as: and

ID is: 4232 Seed is: 1648

Deductions from congruency: evaluating proofs

In the diagram below, D^=71°, CB^A=47°, and DB^C=AC^B=62°.

Prove that BD=CA.

Sarah has already answered the question: her proof is written below. Look carefully at her proof and identify where she has made her mistake.

BD=CA(ΔBDCΔCAB)
Answer:

Sarah's proof is .

Sarah must:

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

What must you do before assuming that two triangles are congruent?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

We cannot use congruency to deduce that BD=CA unless we are certain that the triangles are congruent. Sometimes, we are told in the question that the triangles are congruent. However, this is not the case here. We must first prove that ΔBDCΔCAB before using this fact to answer the question.

Here is the completed proof:

In ΔBDC and ΔCAB:

  1. BC is common
  2. DB^C=AC^B=62° (given)
  3. A^=71° (sum of s in Δ)
    A^=D^

ΔBDCΔCAB (SAA)

BD=CA (ΔBDCΔCAB)


Submit your answer as: and

ID is: 4232 Seed is: 8235

Deductions from congruency: evaluating proofs

In the diagram below, M^=78°, KL^J=50°, and ML^K=JK^L=52°.

Prove that LM=KJ.

Jessica has already answered the question: her proof is written below. But, she has made a mistake! Look carefully at her proof and identify where she has made her mistake.

Line
one In ΔLMK and ΔKJL:
two 1. ML^K=JK^L=52° (given)
three 2. LK is common
four 3. J^=78° (sum of s in Δ)
five J^=M^
six ΔLMKΔKJL
seven LM=KJ (ΔLMKΔKJL)
Answer:

The mistake is on line .

Replace this line with:

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option does not have a reason for the congruency on line six. We must always write the reason why we know that the triangles are congruent: in this case it is SAA.

Line
one In ΔLMK and ΔKJL:
two 1. ML^K=JK^L=52° (given)
three 2. LK is common
four 3. J^=78° (sum of s in Δ)
five J^=M^
six ΔLMKΔKJL (SAA)
seven LM=KJ (ΔLMKΔKJL)

Submit your answer as: and

ID is: 4232 Seed is: 529

Deductions from congruency: evaluating proofs

In the diagram below, R^=75°, SP^Q=46°, and RP^S=QS^P=59°.

Prove that PR=SQ.

Sandile has already answered the question: his proof is written below. But, he has made a mistake! Look carefully at his proof and identify where he has made his mistake.

Line
one In ΔPRS and ΔSQP:
two 1. Q^=75° (sum of s in Δ)
three Q^=R^
four 2. PSR=SPQ (sum of s in Δ)
five 3. RP^S=QS^P=59° (given)
six ΔPRSΔSQP (SAA)
seven PR=SQ (ΔPRSΔSQP)
Answer:

The mistake is on line .

Replace this line with:

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option has a mistake in the congruency "proof".

The mistake is on line four: the angles did not have hats on them. But the bigger problem with this proof is that it does not prove a case for congruency.

This option shows that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.

We have to prove congruency in another way. We can use the fact that PS is common to prove SAA.

Line
one In ΔPRS and ΔSQP:
two 1. Q^=75° (sum of s in Δ)
three Q^=R^
four 2. PS is common
five 3. RP^S=QS^P=59° (given)
six ΔPRSΔSQP (SAA)
seven PR=SQ (ΔPRSΔSQP)

Submit your answer as: and

ID is: 4186 Seed is: 2341

Prove simple congruency

In the diagram below, EB and AC are straight lines that intersect at D. Also, ED^A=90°, ED=BD, and EA=BC.

Prove that ΔEDAΔBDC.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔEDA and ΔBDC:

  1. ED=BD (given)

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that ED=BD, so we have a pair of equal sides.

We were given that ED^A=90°. We can use vertically opposite angles to prove that ED^A=BD^C=90°. So we have a pair of 90° angles.

We were given that EA=BC, so we have another pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔEDA and ΔBDC:

  1. ED=BD (given)
  2. EDA=BDC=90° (vert opp s equal)
  3. EA=BC (given)

ΔEDAΔBDC (90°HS).

NOTE: You may have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the known sides.

Submit your answer as: andandandandand

ID is: 4186 Seed is: 6958

Prove simple congruency

In the diagram below, EC and BD are straight lines that intersect at A. Also, EA^B=90°, EA=CA, and EB=CD.

Prove that ΔEABΔCAD.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔEAB and ΔCAD:

  1. EA=CA (given)

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that EA=CA, so we have a pair of equal sides.

We were given that EA^B=90°. We can use vertically opposite angles to prove that EA^B=CA^D=90°. So we have a pair of 90° angles.

We were given that EB=CD, so we have another pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔEAB and ΔCAD:

  1. EA=CA (given)
  2. EAB=CAD=90° (vert opp s equal)
  3. EB=CD (given)

ΔEABΔCAD (90°HS).

NOTE: You may have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the known sides.

Submit your answer as: andandandandand

ID is: 4186 Seed is: 3188

Prove simple congruency

In the diagram below, QR and SP are straight lines that intersect at T. Also, QT^S=90°, QT=RT, and QS=RP.

Prove that ΔQTSΔRTP.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔQTS and ΔRTP:

  1. QT=RT (given)

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that QT=RT, so we have a pair of equal sides.

We were given that QT^S=90°. We can use vertically opposite angles to prove that QT^S=RT^P=90°. So we have a pair of 90° angles.

We were given that QS=RP, so we have another pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔQTS and ΔRTP:

  1. QT=RT (given)
  2. QTS=RTP=90° (vert opp s equal)
  3. QS=RP (given)

ΔQTSΔRTP (90°HS).

NOTE: You may have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the known sides.

Submit your answer as: andandandandand

ID is: 4178 Seed is: 6876

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. We can see that angle C^ is equal to angle Z^, because they are both 90°. Therefore C and Z must be in the position in the triangles' names.

ΔABCΔZ

We can also see that side BC is equal in length to YZ. Therefore, we must place the letter Y in the same position as B.

ΔABCΔYZ

There is only one vertex (X) left in our second triangle: it will fill in the last empty space.

ΔABCΔXYZ

We can see that we have named the triangle in such a way that the equal angles and equal sides come in the same order for both triangles.

A 90° angle was given in both triangles, together with the hypotenuse and one other side. We use the reason '90°, hypotenuse, side', which is abbreviated to '90°HS'.


Submit your answer as: and

ID is: 4178 Seed is: 6746

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. We can see that angle B^ is equal to angle N^. Therefore, B and N must be in the same position in the triangles' names. In the same way, angle C^ is equal to M^. Therefore C and M must be in the same position in the triangles' names.

ΔABCΔNM

There is only one vertex (L) left in our second triangle: it will fill in the last empty space.

ΔABCΔLNM

We can see that we have named the second triangle in such a way that the equal angles and equal sides come in the same order for both triangles.

One side and two angles were given to be equal, so we use the reason 'side, angle, angle'. This is abbreviated to 'SAA'.


Submit your answer as: and

ID is: 4178 Seed is: 3980

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. This means that the first side named was AB. In our second triangle, the first side we name should be the one that is equal to AB.

Therefore, we will start our second triangle with the letters DE so that side DE will be traced first.

ΔABCΔDE

There is only one vertex (F) left in our second triangle: it will come at the end of the triangle's name.

ΔABCΔDEF

We can see that we have named the second triangle in such a way that the equal sides come in the same order for both triangles.

All three of the pairs of sides were given to be equal, so we use the reason 'side, side, side'. This is abbreviated to 'SSS'.


Submit your answer as: and

ID is: 4234 Seed is: 3145

Prove congruency with calculations

In the diagram below, A^=71°, CB^D=54°, and AB^C=DC^B=55°

Prove that ΔBACΔCDB.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third angle in one of the triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

This option correctly proves that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.

Option B

This option correctly demonstrates one of the four cases for congruency. It uses the given information and sum of angles in a triangle to prove one pair of matching sides and two pairs of matching angles (SAA).

Option C

In Step 2, this options says that the reason why BC^A=DB^C=54° is because the two angles are alternate on parallel lines. But, we have not been told that BDAC, so we cannot use this fact (even if it looks like the lines could be parallel). If we want to use it, we have to prove it first. We can only use the information that we have been given, or that we have proven using our geometry facts.

Option D

In Step 2, this option says that the reason why A^=D^ is because the two triangles are congruent. Since we haven't proved that the triangles are congruent yet, we cannot use the fact that they are congruent. In Geometry we have to know something for certain before we can use it to prove something else.


Submit your answer as:

ID is: 4234 Seed is: 5295

Prove congruency with calculations

In the diagram below, ABCD. Also, CB=10, BA=8, and AD=6.

Prove that ΔCABΔDAB.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third side in one of the right-angled triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

In Step 2, this option says that the reason why BD=10 is because the two triangles are congruent. But, this is what we are trying to prove. We cannot use it as a fact to prove itself.

Option B

In Step 1, this option says that B^ is common. But, a common angle is an angle that is shared by two triangles. In this case, there are two different angles at Point B: CB^A and DB^A. They might be equal, or they might not. They are definitely not common, as each triangle has its own angle at B.

Option C

This proof does not include a reason for the congruency on the last line, so it cannot be the correct choice.

Option D

This option correctly demonstrates one of the four cases for congruency. It uses the given information and the theorem of Pythagoras to prove one pair of matching 90° angles, one pair of hypotenuses, and one pair of matching sides (90°HS).

NOTE: You could also have proved that the triangles were congruent by calculating CA and using SAS.

Submit your answer as:

ID is: 4234 Seed is: 3745

Prove congruency with calculations

In the diagram below, B^=63°, AC^D=58°, and BC^A=DA^C=59°

Prove that ΔCBAΔADC.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third angle in one of the triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

In Step 3, this option says that the reason why B^=D^ is because the two triangles are congruent. Since we haven't proved that the triangles are congruent yet, we cannot use the fact that they are congruent. In Geometry we have to know something for certain before we can use it to prove something else.

Option B

In Step 3, this options says that the reason why CA^B=DC^A=58° is because the two angles are alternate on parallel lines. But, we have not been told that CDBA, so we cannot use this fact (even if it looks like the lines could be parallel). If we want to use it, we have to prove it first. We can only use the information that we have been given, or that we have proven using our geometry facts.

Option C

This option correctly proves that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.

Option D

This option correctly demonstrates one of the four cases for congruency. It uses the given information and sum of angles in a triangle to prove one pair of matching sides and two pairs of matching angles (SAA).


Submit your answer as:

ID is: 4203 Seed is: 1172

Identifying congruency in triangles

Consider ΔXYZ below.

The following triangles all look like they might be congruent to ΔXYZ.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔXYZ?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔXYZ, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 1

Each side of this triangle matches up with one of the sides of ΔXYZ. Therefore, the two triangles are congruent, because of SSS.

Triangle 2

One of the sides of this triangle matches up with a side of ΔXYZ. Two of the angles of this triangle match up with angles in ΔXYZ. Therefore, the two triangles are congruent, because of SAA.

Triangle 3

Two of the sides of this triangle match up to sides of ΔXYZ, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 4

Although this triangle has two sides and one angle which match up to sides and angles in ΔXYZ, the matching angle is not in between the two sides. Therefore, we can't say for sure that this triangle is congruent to ΔXYZ, as SSA is not a case for congruency.


Submit your answer as:

ID is: 4203 Seed is: 8735

Identifying congruency in triangles

Consider ΔABC below.

The following triangles all look like they might be congruent to ΔABC.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔABC?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔABC, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 2

One of the sides of this triangle matches up with a side of ΔABC. Two of the angles of this triangle match up with angles in ΔABC. Therefore, the two triangles are congruent, because of SAA.

Triangle 3

Each side of this triangle matches up with one of the sides of ΔABC. Therefore, the two triangles are congruent, because of SSS.

Triangle 4

Two of the sides of this triangle match up to sides of ΔABC, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 1

Although this triangle has one side and one angle which match up to a side and angle in ΔABC, this is not enough to know for sure that the triangles are congruent. We need at least one more side or angle pair.


Submit your answer as:

ID is: 4203 Seed is: 3211

Identifying congruency in triangles

Consider ΔABC below.

The following triangles all look like they might be congruent to ΔABC.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔABC?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔABC, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 2

Two of the sides of this triangle match up to sides of ΔABC, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 3

Each side of this triangle matches up with one of the sides of ΔABC. Therefore, the two triangles are congruent, because of SSS.

Triangle 4

One of the sides of this triangle matches up with a side of ΔABC. Two of the angles of this triangle match up with angles in ΔABC. Therefore, the two triangles are congruent, because of SAA.

Triangle 1

Although all three of the angles in this triangles are equal to the angles in ΔABC, the two triangles could still have completely different sizes.


Submit your answer as:

ID is: 4202 Seed is: 1790

Congruency in circles

In the diagram below, O is the centre of the circle, and OK=OJ. Also, MJ and KL are straight lines.

Prove that ΔOMKΔOLJ.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔOMK and ΔOLJ:

  1. OK=OJ (given)
  2. MOK=LOJ

ΔOMKΔOLJ

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OK=OJ, so we have a pair of equal sides.

We were also told that MJ and KL are straight lines, so they form vertically opposite angles where they intersect. We can use this fact to prove that MO^K=LO^J . So, we have a pair of equal angles.

Finally, we were told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OM=OL. So, we have another pair of equal sides.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔOMK and ΔOLJ:

  1. OK=OJ (given)
  2. MO^K=LO^J (vert opp s equal)
  3. OM=OL (radii)

ΔOMKΔOLJ (SAS).

NOTE: The reason "radii" is the plural of "radius".

Submit your answer as: andandand

ID is: 4202 Seed is: 8814

Congruency in circles

In the diagram below, O is the centre of the circle, and OJKM.

Prove that ΔOKJΔOMJ.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔOKJ and ΔOMJ:

  1. OJK=OJM=90° (given)

ΔOKJΔOMJ

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OJKM, so we know that OJ^K=OJ^M=90°. So we have a pair of 90° angles.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OK=OM, so we have a pair of equal sides. However, OK is the hypotenuse of ΔOKJ, and OM is the hypotenuse of ΔOMJ, because they are opposite the 90° angles. So we have a pair of equal hypotenuses.

We can see that the line OJ is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔOKJ and ΔOMJ:

  1. OJ^K=OJ^M=90° (given)
  2. OK=OM (radii)
  3. OJ is common

ΔOKJΔOMJ (90°HS).

NOTE: The reason "radii" is the plural of "radius".

Submit your answer as: andandand

ID is: 4202 Seed is: 6291

Congruency in circles

In the diagram below, O is the centre of the circle, and OC=OE. Also, DE and CB are straight lines.

Prove that ΔODCΔOBE.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔODC and ΔOBE:

  1. OC=OE (given)
  2. DOC=BOE

ΔODCΔOBE

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OC=OE, so we have a pair of equal sides.

We were also told that DE and CB are straight lines, so they form vertically opposite angles where they intersect. We can use this fact to prove that DO^C=BO^E . So, we have a pair of equal angles.

Finally, we were told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OD=OB. So, we have another pair of equal sides.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔODC and ΔOBE:

  1. OC=OE (given)
  2. DO^C=BO^E (vert opp s equal)
  3. OD=OB (radii)

ΔODCΔOBE (SAS).

NOTE: The reason "radii" is the plural of "radius".

Submit your answer as: andandand

ID is: 4220 Seed is: 4688

Consequences of congruency

In this diagram, ΔJKLΔRPQ.

Determine the values of a and b, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • a= °
  • b= cm
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔJKLΔRPQ, which means that J^ is the same size as R^. In the same way, K^ is the same size as P^, and L^ is the same size as Q^.

This means that a=78°, because L matches Q. We use the reason (ΔJKLΔRPQ) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, JK is the same length as RP, KL is the same length as PQ, and JL is the same length as RQ.

This means that b=9 cm, because JK is equivalent to RP. We use the reason (ΔJKLΔRPQ) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

ID is: 4220 Seed is: 8931

Consequences of congruency

In this diagram, ΔQPRΔKLJ.

Determine the values of a and b, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • a= °
  • b= m
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔQPRΔKLJ, which means that Q^ is the same size as K^. In the same way, P^ is the same size as L^, and R^ is the same size as J^.

This means that a=58°, because Q matches K. We use the reason (ΔQPRΔKLJ) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, QP is the same length as KL, PR is the same length as LJ, and QR is the same length as KJ.

This means that b=12.26 m, because QP is equivalent to KL. We use the reason (ΔQPRΔKLJ) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

ID is: 4220 Seed is: 5868

Consequences of congruency

In this diagram, ΔABCΔRPQ.

Determine the values of x and y, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • x= °
  • y= m
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔABCΔRPQ, which means that A^ is the same size as R^. In the same way, B^ is the same size as P^, and C^ is the same size as Q^.

This means that x=43°, because B matches P. We use the reason (ΔABCΔRPQ) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, AB is the same length as RP, BC is the same length as PQ, and AC is the same length as RQ.

This means that y=19.71 m, because AB is equivalent to RP. We use the reason (ΔABCΔRPQ) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

ID is: 4201 Seed is: 5592

Deductions from congruency

In the diagram below, O is the centre of the circle, and OCDE. Also, DC=2 units and EO^C=48°.

Determine the size of DO^C.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔODC and ΔOEC:

  1. OCD=OCE=90° (given)
  2. OC is common

ΔODCΔOEC (90°HS)

DOC=
DOC= °

numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the size of DO^C directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the angles.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the size of DO^C directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the angles.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OCDE, so we know that OC^D=OC^E=90°. So we have a pair of 90° angles.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OD=OE, so we have a pair of equal sides. However, OD is the hypotenuse of ΔODC, and OE is the hypotenuse of ΔOEC, because they are opposite the 90° angles. So we have a pair of equal hypotenuses.

We can see that the line OC is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.

Now we can write the proof of the congruency:

In ΔODC and ΔOEC:

  1. OC^D=OC^E=90° (given)
  2. OD=OE (radii)
  3. OC is common

ΔODCΔOEC (90°HS)


STEP: Deduce the size of DO^C
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

DO^C=EO^C(ΔODCΔOEC)DO^C=48 °
NOTE: We only know that DO^C=EO^C because the triangles are congruent. So, we must write "ΔODCΔOEC" as the reason for this statement.

Submit your answer as: andandandand

ID is: 4201 Seed is: 7581

Deductions from congruency

In the diagram below, O is the centre of the circle, and R is the midpoint of PQ (in other words, PR=QR). Also, QO^R=48°.

Determine the size of PO^R.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔOPR and ΔOQR:

  1. PR=QR (given)
  2. OP=OQ (radii)

ΔOPRΔOQR

POR=
POR= °

numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the size of PO^R directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the angles.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the size of PO^R directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the angles.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that PR=QR, so we have a pair of equal sides.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OP=OQ, so we have a pair of equal sides.

We can see that the line OR is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the SSS (side, side, side) case for congruency.

Now we can write the proof of the congruency:

In ΔOPR and ΔOQR:

  1. PR=QR (given)
  2. OP=OQ (radii)
  3. OR is common

ΔOPRΔOQR (SSS)


STEP: Deduce the size of PO^R
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

PO^R=QO^R(ΔOPRΔOQR)PO^R=48 °
NOTE: We only know that PO^R=QO^R because the triangles are congruent. So, we must write "ΔOPRΔOQR" as the reason for this statement.

Submit your answer as: andandandand

ID is: 4201 Seed is: 3831

Deductions from congruency

In the diagram below, O is the centre of the circle, and OC=OE. Also, BE and CD are straight lines, DE=9 units, and C^=83°.

Determine the length of BC.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔOBC and ΔODE:

  1. OC=OE (given)
  2. BOC=DOE (vert opp s equal)

ΔOBCΔODE (SAS)

BC=
BC= units

numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the length of BC directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the sides.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the length of BC directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the sides.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OC=OE, so we have a pair of equal sides.

We were also told that BE and CD are straight lines, so they form vertically opposite angles where they intersect. We can use this fact to prove that BO^C=DO^E . So, we have a pair of equal angles.

Finally, we were told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OB=OD. So, we have another pair of equal sides.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.

Now we can write the proof of the congruency:

In ΔOBC and ΔODE:

  1. OC=OE (given)
  2. BO^C=DO^E (vert opp s equal)
  3. OB=OD (radii)

ΔOBCΔODE (SAS)


STEP: Deduce the length of BC
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

BC=DE(ΔOBEΔOCD)BC=9 units
NOTE: We only know that BC=DE because the triangles are congruent. So, we must write "ΔOBEΔOCD" as the reason for this statement.

Submit your answer as: andandandand

ID is: 4219 Seed is: 5396

Consequences of congruency with calculations

In this diagram, ΔPQRΔACB.

Determine the value of k, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for k. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

k= °

In my working out, I used the following reason(s):

numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching angles will be equal. Can you use a geometry reason to find the size of the angle which matches k?


STEP: Match up the angles of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔPQRΔACB which means that P^ is the same size as A^. In the same way, Q^ is the same size as C^, and R^ is the same size as B^. The matching sides are also equal, but for this question we only need to think about the angles.

This means that k=P^, because A matches P. We use the reason (ΔPQRΔACB) to remind ourselves that the angles are only equal because the triangles are congruent.


STEP: Find the value of the missing angle
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of k in triangle ACB.

But we know that k=P^ and we have enough information to find P^ in triangle PQR, using sum of angles in a triangle.

71°+60°+P^=180°(sum of s in Δ)P^=49°k=49°(ΔPQRΔACB)

Submit your answer as: and

ID is: 4219 Seed is: 175

Consequences of congruency with calculations

In this diagram, ΔQRPΔZYX.

Determine the value of b, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for b. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

b= units

In my working out, I used the following reason(s):

numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching sides will be equal. Can you use a geometry reason to find the length of the side which matches b?


STEP: Match up the sides of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔQRPΔZYX which means that QR is the same length as ZY. In the same way, RP is the same length as YX, and QP is the same length as ZX. The matching angles are also equal, but for this question we only need to think about the sides.

This means that b=PQ, because XZ is equivalent to PQ. We use the reason (ΔQRPΔZYX) to remind ourselves that the sides are only equal because the triangles are congruent.


STEP: Find the value of the missing side
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of b in triangle ZYX.

But we know that b=PQ, and we have enough information to find PQ in triangle QRP, using the theorem of Pythagoras.

RP is the hypotenuse, because it is opposite the 90° angle.
(40)2=(PQ)2+(24)2(Pythagoras)(PQ)2=1600576(PQ)2=1024PQ=32b=32(ΔQRPΔZYX)

Submit your answer as: and

ID is: 4219 Seed is: 6128

Consequences of congruency with calculations

In this diagram, ΔACBΔRQP.

Determine the value of k, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for k. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

k= °

In my working out, I used the following reason(s):

numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching angles will be equal. Can you use a geometry reason to find the size of the angle which matches k?


STEP: Match up the angles of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔACBΔRQP which means that A^ is the same size as R^. In the same way, C^ is the same size as Q^, and B^ is the same size as P^. The matching sides are also equal, but for this question we only need to think about the angles.

This means that k=A^, because R matches A. We use the reason (ΔACBΔRQP) to remind ourselves that the angles are only equal because the triangles are congruent.


STEP: Find the value of the missing angle
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of k in triangle RQP.

But we know that k=A^ and we have enough information to find A^ in triangle ACB, using sum of angles in a triangle.

50°+80°+A^=180°(sum of s in Δ)A^=50°k=50°(ΔACBΔRQP)

Submit your answer as: and

ID is: 4221 Seed is: 6874

Identify congruency in overlapping triangles

Consider the diagram below:

PQ and PR are equal in length. Also, QP^T=PR^S and PT^Q=PS^R.

Identify which triangle is congruent to ΔRPS. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔRPSΔ
string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

We were given PQ=PR.

We were also given that QP^T=PR^S and PT^Q=PS^R.

This means that ΔRPSΔPQT(SAA).

TIP: You must name ΔPQT in that exact order, because that is the order in which the equal angles and sides will match up with ΔRPS.

Submit your answer as: and

ID is: 4221 Seed is: 4020

Identify congruency in overlapping triangles

Consider the diagram below:

AB and AC are equal in length. Also, BA^E=AC^D and AE^B=AD^C.

Identify which triangle is congruent to ΔCAD. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔCADΔ
string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

We were given AB=AC.

We were also given that BA^E=AC^D and AE^B=AD^C.

This means that ΔCADΔABE(SAA).

TIP: You must name ΔABE in that exact order, because that is the order in which the equal angles and sides will match up with ΔCAD.

Submit your answer as: and

ID is: 4221 Seed is: 8845

Identify congruency in overlapping triangles

Consider the diagram below:

JN and NM are equal in length. NK and NL are also equal in length. In addition, JN^K is equal to MN^L, and NL^K is equal to NK^L.

Identify which triangle is congruent to ΔMLN. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔMLNΔ
string
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

We were given JN=NM and KN=NL.

We were also given JN^K=x=MN^L.

This means that ΔMLNΔJKN(SAS).

TIP: You must name ΔJKN in that exact order, because that is the order in which the equal angles and sides will match up with ΔMLN.

Submit your answer as: and

ID is: 4249 Seed is: 341

Deductions from congruency: overlapping triangles

In the diagram below, QTRS. Also, PQ=PR and QP^T=PR^S.

  1. Complete the congruency proof below:

    Answer:

    In ΔPQT and ΔPRS:

    1. QPT=PRS(given)
    2. PQ=PR(given)

    ΔQTPΔ

    string
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔPQT and ΔPRS. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔPQT and ΔPRS:

    1. QP^T=PR^S(given)
    2. PQ=PR(given)
    3. RS^P=QT^P (corresp s;QTRS)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔQTPΔPSR (SAA).


    Submit your answer as: andandand
  2. You are now given that RS=21 and QT=25. Hence, determine the length of TS.

    Answer: TS= units
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    TS is the difference between the length of PS and PT. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine PS and PT
    [−2 points ⇒ 1 / 3 points left]

    PS is made out of PT and TS put together. So PS=PT+TS. If we know PS and PT, we can calculate TS.

    RS=21(given)PT=RS=21(ΔQTPΔPSR)
    QT=25(given)PS=QT=25(ΔQTPΔPSR)

    STEP: Hence, determine TS
    [−1 point ⇒ 0 / 3 points left]
    PS=PT+TS25=21+TSTS=4 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

ID is: 4249 Seed is: 9362

Deductions from congruency: overlapping triangles

In the diagram below, BECD. Also, AB=AC and BA^E=AC^D.

  1. Complete the congruency proof below:

    Answer:

    In ΔABE and ΔACD:

    1. BAE=ACD(given)
    2. AB=AC(given)

    ΔEABΔ

    string
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔABE and ΔACD. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔABE and ΔACD:

    1. BA^E=AC^D(given)
    2. AB=AC(given)
    3. CD^A=BE^A (corresp s;BECD)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔEABΔDCA (SAA).


    Submit your answer as: andandand
  2. You are now given that CD=10 and BE=12. Hence, determine the length of ED.

    Answer: ED= units
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    ED is the difference between the length of AD and AE. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine AD and AE
    [−2 points ⇒ 1 / 3 points left]

    AD is made out of AE and ED put together. So AD=AE+ED. If we know AD and AE, we can calculate ED.

    CD=10(given)AE=CD=10(ΔEABΔDCA)
    BE=12(given)AD=BE=12(ΔEABΔDCA)

    STEP: Hence, determine ED
    [−1 point ⇒ 0 / 3 points left]
    AD=AE+ED12=10+EDED=2 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

ID is: 4249 Seed is: 3543

Deductions from congruency: overlapping triangles

In the diagram below, QTRS. Also, PQ=PR and QP^T=PR^S.

  1. Complete the congruency proof below:

    Answer:

    In ΔPQT and ΔPRS:

    1. QPT=PRS(given)
    2. PQ=PR(given)

    ΔPTQΔ

    string
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔPQT and ΔPRS. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔPQT and ΔPRS:

    1. QP^T=PR^S(given)
    2. PQ=PR(given)
    3. RS^P=QT^P (corresp s;QTRS)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔPTQΔRSP (SAA).


    Submit your answer as: andandand
  2. You are now given that RS=22 and QT=26. Hence, determine the length of TS.

    Answer: TS= units
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    TS is the difference between the length of PS and PT. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine PS and PT
    [−2 points ⇒ 1 / 3 points left]

    PS is made out of PT and TS put together. So PS=PT+TS. If we know PS and PT, we can calculate TS.

    RS=22(given)PT=RS=22(ΔPTQΔRSP)
    QT=26(given)PS=QT=26(ΔPTQΔRSP)

    STEP: Hence, determine TS
    [−1 point ⇒ 0 / 3 points left]
    PS=PT+TS26=22+TSTS=4 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

ID is: 4233 Seed is: 6559

Deductions from congruency: evaluating proofs

In the diagram below, ACBD. Also, BC=13, CA=12, and AD=5.

Prove that BC^A=DC^A.

Emma has already answered the question: her proof is written below. Look carefully at her proof and identify where she has made her mistake.

BC^A=DC^A(ΔBACΔDAC)
Answer:

Emma's proof is .

Emma must:

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

What must you do before assuming that two triangles are congruent?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

We cannot use congruency to deduce that BC^A=DC^A unless we are certain that the triangles are congruent. Sometimes, we are told in the question that the triangles are congruent. However, this is not the case here. We must first prove that ΔBACΔDAC before using this fact to answer the question.

Here is the completed proof:

In ΔBAC and ΔDAC:

  1. BA^C=DA^C=90° (given)
  2. CD2=122+52 (Pythagoras)
    CD=13
    CD=BC
  3. CA is common

ΔBACΔDAC (90°HS)

BC^A=DC^A (ΔBACΔDAC)


Submit your answer as: and

ID is: 4233 Seed is: 677

Deductions from congruency: evaluating proofs

In the diagram below, SQRP. Also, RQ=13, QS=12, and SP=5.

Prove that RQ^S=PQ^S.

Nthabiseng has already answered the question: her proof is written below. But, she has made a mistake! Look carefully at her proof and identify where she has made her mistake.

Line
one In ΔRSQ and ΔPSQ:
two 1. QP2=122+52 (Pythagoras)
three QP=13
four QP=RQ
five 2. QS is common
six 3. RS^Q=PS^Q=90° (given)
seven ΔRSQΔPSQ
eight RQ^S=PQ^S (ΔRSQΔPSQ)
Answer:

The mistake is on line .

Replace this line with:

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option does not have a reason for the congruency on line seven. We must always write the reason why we know that the triangles are congruent: in this case it is 90°HS.

Line
one In ΔRSQ and ΔPSQ:
two 1. QP2=122+52 (Pythagoras)
three QP=13
four QP=RQ
five 2. QS is common
six 3. RS^Q=PS^Q=90° (given)
seven ΔRSQΔPSQ (90°HS)
eight RQ^S=PQ^S (ΔRSQΔPSQ)

Submit your answer as: and

ID is: 4233 Seed is: 226

Deductions from congruency: evaluating proofs

In the diagram below, QPSR. Also, SP=34, PQ=30, and QR=16.

Prove that SP^Q=RP^Q.

Chris has already answered the question: his proof is written below. Look carefully at his proof and identify where he has made his mistake.

SP^Q=RP^Q(ΔSQPΔRQP)
Answer:

Chris's proof is .

Chris must:

2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

What must you do before assuming that two triangles are congruent?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

We cannot use congruency to deduce that SP^Q=RP^Q unless we are certain that the triangles are congruent. Sometimes, we are told in the question that the triangles are congruent. However, this is not the case here. We must first prove that ΔSQPΔRQP before using this fact to answer the question.

Here is the completed proof:

In ΔSQP and ΔRQP:

  1. PQ is common
  2. SQ^P=RQ^P=90° (given)
  3. PR2=302+162 (Pythagoras)
    PR=34
    PR=SP

ΔSQPΔRQP (90°HS)

SP^Q=RP^Q (ΔSQPΔRQP)


Submit your answer as: and

ID is: 4248 Seed is: 238

Deductions from congruency: overlapping triangles

In the diagram below, ABDC. Also, AB^C=DC^B=90° and BD^C=DC^A=x.

  1. Chinyelu needs to prove that ΔABCΔDCB. She has already started and her incomplete proof is written below:

    In ΔABC and ΔDCB:

    1. AB^C=DC^B=90° (given)
    2. BC is common
      ...
    Answer:

    How should Chinyelu complete her proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔABCΔDCB. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. ABDC (given)
    ΔABCΔDCB (SAS)

    AB is parallel to DC. But, this does not necessarily tell us anything about the length of AB or DC, becuase parallel lines do not have to be equal in length. This option does not prove that another pair of sides are equal.
    B ✔

    3. D^=x (given)
    Also, A^=AC^D=x (alt s; ABDC)
    A^=D^ (both equal to x)
    ΔABCΔDCB (SAA)

    This option uses geometry reasons correctly to prove that A^=D^. This is a pair of matching angles in both triangles. So this proves that ΔABCΔDCB, using SAA.
    C ✘

    3. AB^D=x (alt s; ABDC)
    AB^D=DC^A (both equal to x)
    ΔABCΔDCB (SAA)

    This option correctly shows that AB^D is equal to DC^A. But, these are not angles in the triangles that we are trying to prove are congruent.
    D ✘

    3. AE^B=DE^C (vert opp s)
    ΔABCΔDCB (SAA)

    AE^B is equal to DE^C, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔABC and ΔDCB:

    1. AB^C=DC^B=90° (given)
    2. BC is common
    3. D^=x (given)
      Also, A^=AC^D=x (alt s; ABDC)
      A^=D^ (both equal to x)

    ΔABCΔDCB (SAA)


    Submit your answer as:
  2. You are now given that BC=8 and CD=4.

    Hence, determine the length of AC.

    INSTRUCTION: Round off your answer to two decimal places.
    Answer: AC= units
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches AC? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate DB
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate AC directly. But, in ΔDCB we can calculate DB using the theorem of Pythagoras. Then we will use the fact that AC and DB are matching sides in congruent triangles, to find AC.

    DB2=DC2+BC2(Pythagoras)DB2=42+82DB2=80DB=80=8.94427...8.94 units

    STEP: Hence, determine AC using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    AC=DB(ΔABCΔDCB)AC=8.94 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

ID is: 4248 Seed is: 3515

Deductions from congruency: overlapping triangles

In the diagram below, TR=TQ and PQ^R=SR^Q=90°.

  1. Siyabonga needs to prove that ΔPQRΔSRQ. He has already started and his incomplete proof is written below:

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
      ...
    Answer:

    How should Siyabonga complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔPQRΔSRQ. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. TQ=TR (given)
    ΔPQRΔSRQ (90°HS)

    TQ is equal to TR. But, these are not sides in the triangles that we are trying to prove are congruent. (They are only parts of the sides in those triangles, which is not enough.)
    B ✘

    3. PR is the hypotenuse of ΔPQR
    Also, SQ is the hypotenuse of ΔSRQ.
    PR=SQ (hypotenuse with common base)
    ΔPQRΔSRQ (90°HS)

    PR and SQ are hypotenuses of ΔPQR and ΔSRQ respectively. But, this does not mean that they have to be equal in length. Hypotenuse with common base does not refer to any of our geometry theorems. It is just a made up reason. So, this option does not prove that PR=SQ.
    C ✘

    3. PT^Q=ST^R (vert opp s)
    ΔPQRΔSRQ (SAA)

    PT^Q is equal to ST^R, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.
    D ✔

    3. TQ^R=TR^Q (s opp equal sides)
    ΔPQRΔSRQ (SAA)

    This option uses geometry reasons correctly to prove that TQ^R=TR^Q. This is a pair of matching angles in both triangles. So this proves that ΔPQRΔSRQ, using SAA.

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
    3. TQ^R=TR^Q (s opp equal sides)

    ΔPQRΔSRQ (SAA)


    Submit your answer as:
  2. You are now given that QR=8 and RS=4.

    Hence, determine the length of PR.

    INSTRUCTION: Round off your answer to two decimal places.
    Answer: PR= units
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches PR? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate SQ
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate PR directly. But, in ΔSRQ we can calculate SQ using the theorem of Pythagoras. Then we will use the fact that PR and SQ are matching sides in congruent triangles, to find PR.

    SQ2=SR2+QR2(Pythagoras)SQ2=42+82SQ2=80SQ=80=8.94427...8.94 units

    STEP: Hence, determine PR using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    PR=SQ(ΔPQRΔSRQ)PR=8.94 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

ID is: 4248 Seed is: 66

Deductions from congruency: overlapping triangles

In the diagram below, ABDC. Also, AB^C=DC^B=90° and BD^C=DC^A=x.

  1. Suleiman needs to prove that ΔABCΔDCB. He has already started and his incomplete proof is written below:

    In ΔABC and ΔDCB:

    1. AB^C=DC^B=90° (given)
    2. BC is common
      ...
    Answer:

    How should Suleiman complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔABCΔDCB. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. AB^D=x (alt s; ABDC)
    AB^D=DC^A (both equal to x)
    ΔABCΔDCB (SAA)

    This option correctly shows that AB^D is equal to DC^A. But, these are not angles in the triangles that we are trying to prove are congruent.
    B ✔

    3. D^=x (given)
    Also, A^=AC^D=x (alt s; ABDC)
    A^=D^ (both equal to x)
    ΔABCΔDCB (SAA)

    This option uses geometry reasons correctly to prove that A^=D^. This is a pair of matching angles in both triangles. So this proves that ΔABCΔDCB, using SAA.
    C ✘

    3. AC is the hypotenuse of ΔABC
    Also, DB is the hypotenuse of ΔDCB.
    AC=DB (hypotenuse with common base)
    ΔABCΔDCB (90°HS)

    AC and DB are hypotenuses of ΔABC and ΔDCB respectively. But, this does not mean that they have to be equal in length. Hypotenuse with common base does not refer to any of our geometry theorems. It is just a made up reason. So, this option does not prove that AC=DB.
    D ✘

    3. AE^B=DE^C (vert opp s)
    ΔABCΔDCB (SAA)

    AE^B is equal to DE^C, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔABC and ΔDCB:

    1. AB^C=DC^B=90° (given)
    2. BC is common
    3. D^=x (given)
      Also, A^=AC^D=x (alt s; ABDC)
      A^=D^ (both equal to x)

    ΔABCΔDCB (SAA)


    Submit your answer as:
  2. You are now given that BC=6 and CD=8.

    Hence, determine the length of AC.

    Answer: AC= units
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches AC? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate DB
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate AC directly. But, in ΔDCB we can calculate DB using the theorem of Pythagoras. Then we will use the fact that AC and DB are matching sides in congruent triangles, to find AC.

    DB2=DC2+BC2(Pythagoras)DB2=82+62DB2=100DB=100=10 units

    STEP: Hence, determine AC using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    AC=DB(ΔABCΔDCB)AC=10.0 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

3. Properties of quadrilaterals


ID is: 4282 Seed is: 5364

Apply diagonal properties of a rhombus

In the diagram below, PQRS is a rhombus. Also, PS= 10 and SM= 8.

Determine, with reasons, the length of PR.

INSTRUCTION: Answer this question by completing the steps below.
Answer: PM^S= °
PM= units
Hence, PR= units
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

PR=PM+MR. In a rhombus, PM=MR. So, if we know PM, we can work out PR. We are going to use the theorem of Pythagoras in ΔPMS to determine PM. To do this, we will use rhombus diagonal properties to prove that ΔPMS is right-angled.


STEP: Show that ΔPMS is right-angled
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus intersect at 90°.

PM^S=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of PM
[−2 points ⇒ 2 / 6 points left]

We have demonstrated that ΔPMS is right-angled. Now, we can use the theorem of Pythagoras to work out PM.

PM2+SM2=PS2 (Pythagoras)

PM2+82=102PM2+64=100PM2=36PM=36PM=6 units

STEP: Apply diagonal properties to determine PR
[−2 points ⇒ 0 / 6 points left]

The diagonals of a rhombus bisect each other.

We know that PM= 6 so:

PM=MR (diags of rhombus)
MR= 6 units

Also, PR=PM+MR (from the diagram)
PR= 12 units


Submit your answer as: andandandandand

ID is: 4282 Seed is: 4149

Apply diagonal properties of a rhombus

In the diagram below, ABCD is a rhombus. Also, AD=25 and AC=30.

Determine, with reasons, the length of DE.

INSTRUCTION: Answer this question by completing the steps below.
Answer: AE= units
Also, AE^D= °
Hence, DE= units
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

We are going to use the theorem of Pythagoras in ΔAED to determine AE. To do this, we will use rhombus diagonal properties to determine the necessary sides, and to prove that ΔAED is right-angled.


STEP: Apply diagonal properties to determine AE
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus bisect each other.

AC=30 (given)
And, AE=EC (diags of rhombus)
AE= 15 units


STEP: Show that ΔAED is right-angled
[−2 points ⇒ 2 / 6 points left]
The diagonals of a rhombus intersect at 90°.

AE^D=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of DE
[−2 points ⇒ 0 / 6 points left]

We have shown that ΔAED is right-angled. Now, we can use the theorem of Pythagoras to work out DE.

DE2+AE2=AD2 (Pythagoras)

DE2+152=252DE2+225=625DE2=400DE=400DE=20 units

Submit your answer as: andandandandand

ID is: 4282 Seed is: 326

Apply diagonal properties of a rhombus

In the diagram below, ABCD is a rhombus. Also, AD= 13 and AE= 5.

Determine, with reasons, the length of DB.

INSTRUCTION: Answer this question by completing the steps below.
Answer: AE^D= °
DE= units
Hence, DB= units
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

DB=DE+EB. In a rhombus, DE=EB. So, if we know DE, we can work out DB. We are going to use the theorem of Pythagoras in ΔAED to determine DE. To do this, we will use rhombus diagonal properties to prove that ΔAED is right-angled.


STEP: Show that ΔAED is right-angled
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus intersect at 90°.

AE^D=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of DE
[−2 points ⇒ 2 / 6 points left]

We have demonstrated that ΔAED is right-angled. Now, we can use the theorem of Pythagoras to work out DE.

DE2+AE2=AD2 (Pythagoras)

DE2+52=132DE2+25=169DE2=144DE=144DE=12 units

STEP: Apply diagonal properties to determine DB
[−2 points ⇒ 0 / 6 points left]

The diagonals of a rhombus bisect each other.

We know that DE= 12 so:

DE=EB (diags of rhombus)
EB= 12 units

Also, DB=DE+EB (from the diagram)
DB= 24 units


Submit your answer as: andandandandand

ID is: 4214 Seed is: 972

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

In parallelograms, the opposite sides are equal in length.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

Even though it is not part of the definition, the opposite sides of a parallelogram are always the same length.

So, the statement is always true.


Submit your answer as:

ID is: 4214 Seed is: 5435

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

Rectangles have two pairs of opposite sides that are parallel.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

The opposite sides of a rectangle are always parallel, because a rectangle is a special type of parallelogram.

So, the statement is always true.


Submit your answer as:

ID is: 4214 Seed is: 1428

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

Squares have two pairs of opposite sides that are parallel.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

The opposite sides of a square are always parallel, because a square is a special type of parallelogram.

So, the statement is always true.


Submit your answer as:

ID is: 4250 Seed is: 5688

Identify all diagonal properties

Think about the diagonals of a rhombus.

Select whether each of the following statements is true or false for all rhombuses.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard rhombus and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

Rhombuses have three diagonal properties that we can prove.

The diagonals of a kite intersect at 90°. A rhombus is a special type of kite, so the diagonals of a rhombus must also intersect at 90°.

The diagonals of a parallelogram bisect each other. A rhombus is a special type of parallelogram, so the diagonals of a rhombus must also bisect each other.

When we draw in the diagonals, we create several pairs of congruent triangles. In particular ΔPTS, ΔRTS, ΔRTQ and ΔPTQ are all congruent to each other. (Think about how you would prove this!)

This means that:

  • PS^T=RS^T ,
  • SR^T=QR^T ,
  • RQ^T=PQ^T , and
  • QP^T=SP^T .

The corner angles are all cut into two equal parts by the diagonals, so the diagonals bisect the corner angles.

These are the only properties that we can prove for certain about the diagonals of all rhombuses. The following table shows the correct answers:

Diagonal property Definitely true for any rhombus?
The diagonals bisect each other. True
Both diagonals are the same length. False
The diagonals intersect at 90°. True
The diagonals bisect the corner angles. True

Submit your answer as: andandand

ID is: 4250 Seed is: 5741

Identify all diagonal properties

Think about the diagonals of a square.

Select whether each of the following statements is true or false for all squares.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard square and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

A square is a special type of rectangle and a special type of rhombus. So it will have all of the diagonal properties of rectangles and of rhombuses.

The following table shows the correct answers:

Diagonal property Definitely true for any square?
The diagonals bisect each other. True
Both diagonals are the same length. True
The diagonals intersect at 90°. True
The diagonals bisect the corner angles. True

Submit your answer as: andandand

ID is: 4250 Seed is: 3628

Identify all diagonal properties

Think about the diagonals of a kite.

Select whether each of the following statements is true or false for all kites.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard kite and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

When we draw in the diagonals, we create several pairs of congruent triangles, including ΔABEΔADE. (This is a bit harder to prove - but you should still try!) This means that E1 and E2 are equal. We also know that E1 and E2 add up to 180°. So, they must both be 90°. So, the diagonals intersect at 90°.

NOTE: One diagonal of a kite, BD, is bisected by the other. But AC is not bisected. So, we do not say that the diagonals bisect each other in a kite.

This is the only property that we can prove for certain about the diagonals of all kites. The following table shows the correct answers:

Diagonal property Definitely true for any kite?
The diagonals bisect each other. False
Both diagonals are the same length. False
The diagonals intersect at 90°. True
The diagonals bisect the corner angles. False

Submit your answer as: andandand

ID is: 1654 Seed is: 6367

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    Note that the two pairs of sides are equal, as indicated by the m and n. In addition, the angle between those two sides are marked as equal.

    Therefore, these two triangles are congruent, and the reason is SAS.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔCABΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does A^ correspond to angle D^,E^, or F^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    The first triangle can be named in any order - we are given ΔCAB. The second triangle needs to be named so that we can read off the corresponding angles and sides: we need to ensure that the corresponding sides and angles match up.

    For example, vertex C corresponds to vertex F. Therefore, the answer must start with F. Now think about which vertex corresponds to vertex A to figure out which letter should be second in the answer.

    The correct choice to complete the statement is ΔFDE.


    Submit your answer as:

ID is: 1654 Seed is: 2637

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    We are given information about two angles and one side: there are two pairs of corresponding angles that are equal (indicated by the dots and the stars), and one pair of sides which are the same length.

    Therefore, these two triangles are congruent, and the reason is AAS.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔVUWΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does U^ correspond to angle X^,Y^, or Z^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    The first triangle can be named in any order - we are given ΔVUW. The second triangle needs to be named so that we can read off the corresponding angles and sides: we need to ensure that the corresponding sides and angles match up.

    For example, vertex V corresponds to vertex Y. Therefore, the answer must start with Y. Now think about which vertex corresponds to vertex U to figure out which letter should be second in the answer.

    The correct choice to complete the statement is ΔYXZ.


    Submit your answer as:

ID is: 1654 Seed is: 4741

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    The sides of both triangles are labelled with m, n and p. This means that there are three pairs of corresponding and equal sides.

    Therefore, these two triangles are congruent, and the reason is SSS.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔVUWΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does U^ correspond to angle X^,Y^, or Z^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    The first triangle can be named in any order - we are given ΔVUW. The second triangle needs to be named so that we can read off the corresponding angles and sides: we need to ensure that the corresponding sides and angles match up.

    For example, vertex V corresponds to vertex Y. Therefore, the answer must start with Y. Now think about which vertex corresponds to vertex U to figure out which letter should be second in the answer.

    The correct choice to complete the statement is ΔYXZ.


    Submit your answer as:

ID is: 1865 Seed is: 636

Calculate angles between parallel lines

The diagram represents transversal line FG that intersects with the straight lines BC and DE respectively. BC DE.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=82°
  2. y=82°
  3. z=98°

Submit your answer as: andand

ID is: 1865 Seed is: 4969

Calculate angles between parallel lines

The diagram represents transversal line LM that intersects with the straight lines GH and JK respectively. GH JK.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=78°
  2. y=78°
  3. z=102°

Submit your answer as: andand

ID is: 1865 Seed is: 1784

Calculate angles between parallel lines

The diagram represents transversal line EF that intersects with the straight lines AB and CD respectively. AB CD.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=81°
  2. y=81°
  3. z=99°

Submit your answer as: andand

ID is: 1670 Seed is: 6143

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral MNOP with MPNO and MP=NO. Diagonal NP is shown with a dashed line. M=y and O=74° ; MPN=x and NPO=57°.

  1. The steps and reasons below prove that MNOP is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO Given
    ΔMNP=ΔOPN Side-Angle-Side for congruent triangles (SAS)
    MNPO MNP & OPN are equal alternate interior angles
    MNOP is a parallelogram
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side MP is parallel to side NO. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides MN and PO are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that MP is parallel to NO. But it takes six steps to first prove that MN and PO are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles MNP and OPN are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    MPN=PNO Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO Given
    ΔMNP=ΔOPN Side-Angle-Side for congruent triangles (SAS)
    MNP=OPN Corresponding angles in congruent triangles
    MNPO MNP & OPN are equal alternate interior angles
    MNOP is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    MNOP is a parallelogram. Opposite angles of a parallelogram are equal. So M is equal to O.

    Therefore, y=74°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle MNP. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle MNP.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    M+MPN+MNP=180°

    From Question 2 we know that y=74° (this is labelled in the diagram above). And because of alternate interior angles we also know that MNP=57°. Putting all of this together:

    M+MPN+MNP=180°74°+x+57°=180°x=180°74°57°x=49°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    74°+74°+2(57°+x)=360°
    If you solve this you will also get the answer x=49°.

    The measure of angle x is 49°.


    Submit your answer as:

ID is: 1670 Seed is: 5538

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral ABCD with ADBC and AD=BC. Diagonal BD is shown with a dashed line. A=y and C=76° ; ADB=45° and BDC=x.

  1. The steps and reasons below prove that ABCD is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    Alternate interior angles are equal if ADBC
    Common side (in triangles ABD & CDB)
    AD=BC
    ΔABD=ΔCDB
    ABD=CDB Corresponding angles in congruent triangles
    ABDC ABD & CDB are equal alternate interior angles
    ABCD is a parallelogram Definition of a parallelogram (opposite sides parallel)
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side AD is parallel to side BC. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides AB and DC are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that AD is parallel to BC. But it takes six steps to first prove that AB and DC are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles ABD and CDB are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    ADB=DBC Alternate interior angles are equal if ADBC
    BD=DB Common side (in triangles ABD & CDB)
    AD=BC Given
    ΔABD=ΔCDB Side-Angle-Side for congruent triangles (SAS)
    ABD=CDB Corresponding angles in congruent triangles
    ABDC ABD & CDB are equal alternate interior angles
    ABCD is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    ABCD is a parallelogram. Opposite angles of a parallelogram are equal. So A is equal to C.

    Therefore, y=76°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle CBD. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle CBD.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    C+CDB+CBD=180°

    From Question 2 we know that y=76° (this is labelled in the diagram above). And because of alternate interior angles we also know that CBD=45°. Putting all of this together:

    C+CDB+CBD=180°76°+x+45°=180°x=180°76°45°x=59°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    76°+76°+2(45°+x)=360°
    If you solve this you will also get the answer x=59°.

    The measure of angle x is 59°.


    Submit your answer as:

ID is: 1670 Seed is: 685

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral MNOP with MPNO and MP=NO. Diagonal NP is shown with a dashed line. M=y and O=76° ; MPN=67° and NPO=x.

  1. The steps and reasons below prove that MNOP is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    NP=PN
    MP=NO Given
    ΔMNP=ΔOPN Side-Angle-Side for congruent triangles (SAS)
    Corresponding angles in congruent triangles
    MNPO MNP & OPN are equal alternate interior angles
    MNOP is a parallelogram Definition of a parallelogram (opposite sides parallel)
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side MP is parallel to side NO. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides MN and PO are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that MP is parallel to NO. But it takes six steps to first prove that MN and PO are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles MNP and OPN are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    MPN=PNO Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO Given
    ΔMNP=ΔOPN Side-Angle-Side for congruent triangles (SAS)
    MNP=OPN Corresponding angles in congruent triangles
    MNPO MNP & OPN are equal alternate interior angles
    MNOP is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    MNOP is a parallelogram. Opposite angles of a parallelogram are equal. So M is equal to O.

    Therefore, y=76°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle ONP. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle ONP.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    O+OPN+ONP=180°

    From Question 2 we know that y=76° (this is labelled in the diagram above). And because of alternate interior angles we also know that ONP=67°. Putting all of this together:

    O+OPN+ONP=180°76°+x+67°=180°x=180°76°67°x=37°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    76°+76°+2(67°+x)=360°
    If you solve this you will also get the answer x=37°.

    The measure of angle x is 37°.


    Submit your answer as:

ID is: 4251 Seed is: 4170

Choose one correct diagonal property

PQRS below is a parallelogram.

Answer:

Which of the following statements is definitely true about the diagonals of PQRS?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the parallelogram to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

When we draw in the diagonals, we create several pairs of congruent triangles. In particular, ΔPTQΔRTS. (Think about how you would prove this!) So, PT=RT and TS=TQ. In other words, the diagonals bisect each other.

NOTE: Bisect means to cut into two equal parts.

This is the only property that we can prove for certain about the diagonals of all parallelograms.

Diagonal property Definitely true for any parallelogram?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Only D is definitely true for all parallelograms.


Submit your answer as:

ID is: 4251 Seed is: 7788

Choose one correct diagonal property

ABCD below is a parallelogram.

Answer:

Which of the following statements is definitely true about the diagonals of ABCD?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the parallelogram to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

When we draw in the diagonals, we create several pairs of congruent triangles. In particular, ΔAEBΔCED. (Think about how you would prove this!) So, AE=CE and ED=EB. In other words, the diagonals bisect each other.

NOTE: Bisect means to cut into two equal parts.

This is the only property that we can prove for certain about the diagonals of all parallelograms.

Diagonal property Definitely true for any parallelogram?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Only B is definitely true for all parallelograms.


Submit your answer as:

ID is: 4251 Seed is: 4418

Choose one correct diagonal property

PQRS below is a kite.

Answer:

Which of the following statements is definitely true about the diagonals of PQRS?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the kite to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

When we draw in the diagonals, we create several pairs of congruent triangles, including ΔPQTΔPST. (This is a bit harder to prove - but you should still try!) This means that T1 and T2 are equal. We also know T1 and T2 add up to 180°, so they must both be 90°. This means the diagonals intersect at 90°.

This is the only property that we can prove for certain about the diagonals of all kites.

Diagonal property Definitely true for any kite?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Only A is definitely true for all kites.

NOTE: One diagonal of a kite, QS, is bisected by the other. But PR is not bisected. So, we do not say that the diagonals bisect each other in a kite.

Submit your answer as:

ID is: 4253 Seed is: 4321

Apply diagonal properties

In the diagram below, AB^E=21° and AD^E=102°.

Answer the following questions:

  1. What type of quadrilateral is ABCD? (Give the most specific name.)
  2. Determine the size of ED^C.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. ABCD is a .
  2. ED^C= °
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of ABCD. How can these help you to determine ED^C?


STEP: Identify ABCD
[−1 point ⇒ 2 / 3 points left]

The most specific name for ABCD is a parallelogram. Even though ABCD is a type of trapezium (for example), this is not the most specific name for ABCD.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a parallelogram to determine ED^C
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a parallelogram:

Diagonal property Definitely true for a parallelogram?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a parallelogram do not necessarily bisect the corner angles, so we cannot assume that ED^C is equal to AD^E. In fact, we must use the fact that alternate angles on parallel lines are equal to deduce that ED^C = AB^E = 21°.



Submit your answer as: and

ID is: 4253 Seed is: 5439

Apply diagonal properties

In the diagram below, PT=13 and PR^S=30°.

Answer the following questions:

  1. What type of quadrilateral is PQRS? (Give the most specific name.)
  2. Determine the size of PR^Q.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. PQRS is a .
  2. PR^Q= °
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of PQRS. How can these help you to determine PR^Q?


STEP: Identify PQRS
[−1 point ⇒ 2 / 3 points left]

The most specific name for PQRS is a rectangle. Even though PQRS is a type of parallelogram (for example), this is not the most specific name for PQRS.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a rectangle to determine PR^Q
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a rectangle:

Diagonal property Definitely true for a rectangle?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a rectangle do not necessarily bisect the corner angles. So, we cannot assume that PR^Q = PR^S. Instead, we must use the fact that QR^S=90°:

QR^S=90° and PR^S =30° (both given)
Also,PR^Q+PR^S=QR^S (from the diagram)

PR^Q+30°=90°PR^Q=60°

Submit your answer as: and

ID is: 4253 Seed is: 8698

Apply diagonal properties

In the diagram below, TS^R=32° and TP^S=58°.

Answer the following questions:

  1. What type of quadrilateral is PQRS? (Give the most specific name.)
  2. Determine the size of PT^S.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. PQRS is a .
  2. PT^S= °
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of PQRS. How can these help you to determine PT^S?


STEP: Identify PQRS
[−1 point ⇒ 2 / 3 points left]

The most specific name for PQRS is a rhombus. Even though PQRS is a type of parallelogram (for example), this is not the most specific name for PQRS.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a rhombus to determine PT^S
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a rhombus:

Diagonal property Definitely true for a rhombus?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a rhombus intersect at 90°. Therefore, PT^S = 90°.



Submit your answer as: and

ID is: 1858 Seed is: 6807

Congruent triangles

In the diagram below, ΔMNPΔMQP. Also, MPQN while NP=10 and MP=8.

  1. Calculate the value of x.
  2. Determine the length of QN.
Answer:
  1. x= units
  2. QN= units
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side MQ.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that MPQN, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle MQP, but you can use either one, because they are congruent.

In ΔMQP:QP=10(ΔMNPΔMQP)(10)2=x2+(8)2(Pythagoras)100=x2+6410064=x2±36=x6=x

The length of MQ is 6 units.


STEP: Find the length of QN
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment QN. This segment is twice as long as x, which we calculated above. So QN=2(6)=12 units.

The correct answers are:

  1. The length of side x is 6 units.
  2. The length of QN is 12 units.

Submit your answer as: and

ID is: 1858 Seed is: 8337

Congruent triangles

In the diagram below, ΔABCΔADC. Also, ACDB while BC=30 and AC=24.

  1. Calculate the value of x.
  2. Determine the length of DB.
Answer:
  1. x= units
  2. DB= units
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side AD.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that ACDB, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle ADC, but you can use either one, because they are congruent.

In ΔADC:DC=30(ΔABCΔADC)(30)2=x2+(24)2(Pythagoras)900=x2+576900576=x2±324=x18=x

The length of AD is 18 units.


STEP: Find the length of DB
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment DB. This segment is twice as long as x, which we calculated above. So DB=2(18)=36 units.

The correct answers are:

  1. The length of side x is 18 units.
  2. The length of DB is 36 units.

Submit your answer as: and

ID is: 1858 Seed is: 873

Congruent triangles

In the diagram below, ΔABCΔADC. Also, ACDB while BC=10 and AC=8.

  1. Calculate the value of x.
  2. Determine the length of DB.
Answer:
  1. x= units
  2. DB= units
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side AD.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that ACDB, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle ADC, but you can use either one, because they are congruent.

In ΔADC:DC=10(ΔABCΔADC)(10)2=x2+(8)2(Pythagoras)100=x2+6410064=x2±36=x6=x

The length of AD is 6 units.


STEP: Find the length of DB
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment DB. This segment is twice as long as x, which we calculated above. So DB=2(6)=12 units.

The correct answers are:

  1. The length of side x is 6 units.
  2. The length of DB is 12 units.

Submit your answer as: and

ID is: 4182 Seed is: 6904

Definitions of quadrilaterals

  1. Select the most correct definition for a parallelogram.

    Answer: A parallelogram is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • "Tilted rectangle" is not a useful way to think about a parallelogram. It suggests that a parallelogram is a type of rectangle, but this is not true.
    • A quadrilateral with at least one pair of opposite sides parallel is a trapezium.
    • A quadrilateral with two pairs of adjacent sides equal is a kite.
    • A parallelogram is a quadrilateral with two pairs of opposite sides parallel.

    This is a diagram of a parallelogram with the definition properties drawn in:


    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a parallelogram (plus some more), so it is a special type of parallelogram.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a parallelogram, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a parallelogram
    [−1 point ⇒ 0 / 1 points left]

    A square has two pairs of opposite sides parallel. This means that a square meets the definition of a parallelogram, so we say that it is a special type of parallelogram.

    NOTE: It does not matter that a square has extra properties: what matters is that it has enough properties to be called a parallelogram.

    Submit your answer as:

ID is: 4182 Seed is: 4030

Definitions of quadrilaterals

  1. Select the most correct definition for a kite.

    Answer: A kite is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • A kite is a quadrilateral with two pairs of adjacent sides equal.
    • A quadrilateral with two pairs of opposite sides parallel is a parallelogram.
    • We do not refer to any quadrilaterals as "diamonds". This is an informal name. As we progress further in mathematics, we need to start to use the correct terms.
    • A quadrilateral with two pairs of opposite sides parallel and all corners 90° is a rectangle.

    This is a diagram of a kite with the definition properties drawn in:


    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a kite (plus some more), so it is a special type of kite.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a kite, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a kite
    [−1 point ⇒ 0 / 1 points left]

    A square has two pairs of adjacent sides equal. This means that a square meets the definition of a kite, so we say that it is a special type of kite.

    NOTE: It does not matter that a square has extra properties: what matters is that it has enough properties to be called a kite.

    Submit your answer as:

ID is: 4182 Seed is: 8567

Definitions of quadrilaterals

  1. Select the most correct definition for a rhombus.

    Answer: A rhombus is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • A rhombus is a quadrilateral with two pairs of opposite sides parallel and adjacent sides equal.
    • "Tilted square" is not a useful way to think about a rhombus. It suggests that a rhombus is a type of square, but this is not true.
    • A quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90° is a square.
    • A quadrilateral with two pairs of opposite sides parallel and all corners 90° is a rectangle.

    This is a diagram of a rhombus with the definition properties drawn in:


    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a rhombus (plus some more), so it is a special type of rhombus.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a rhombus, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a rhombus
    [−1 point ⇒ 0 / 1 points left]

    A square has two pairs of opposite sides parallel and adjacent sides equal. This means that a square meets the definition of a rhombus, so we say that it is a special type of rhombus.

    NOTE: It does not matter that a square has extra properties: what matters is that it has enough properties to be called a rhombus.

    Submit your answer as:

ID is: 1853 Seed is: 2170

Opposites of a parallelogram

Answer the following questions about the parallelogram XWVU. The parallelogram has the following sides and angles:

X^=63°, W^=x, V^=y, and U^=117°. The sides are XW=w, WV=4, VU=7, and XU=4.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=117°
  • y=63°
  • w=7units

Submit your answer as: andand

ID is: 1853 Seed is: 1483

Opposites of a parallelogram

Answer the following questions about the parallelogram XWVU. The parallelogram has the following sides and angles:

X^=y, W^=x, V^=39°, and U^=141°. The sides are XW=6, WV=w, VU=6, and XU=6.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=141°
  • y=39°
  • w=6units

Submit your answer as: andand

ID is: 1853 Seed is: 6630

Opposites of a parallelogram

Answer the following questions about the parallelogram XWVU. The parallelogram has the following sides and angles:

X^=74°, W^=y, V^=x, and U^=106°. The sides are XW=5, WV=7, VU=5, and XU=w.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=74°
  • y=106°
  • w=7units

Submit your answer as: andand

ID is: 4216 Seed is: 7555

Identifying quadrilaterals

Which of the following are kites?

A B
C D
Answer: The kites are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a kites? They could have extra properties as well - that does not mean they are not kites.


STEP: Identify the quadrilaterals that meet the definition of a kite
[−2 points ⇒ 0 / 2 points left]
A kite is a quadrilateral with two pairs of equal adjacent sides.

We can easily recognise Quadrilateral D as a kite. However, this is not the only type of kite in the options.

Quadrilaterals A and C each have two pairs of equal adjacent sides. So, A and C are also types of kites. This is true even though the exact name for A is a square, and the exact name for C is a rhombus (because they have a few extra properties too).


Submit your answer as:

ID is: 4216 Seed is: 4032

Identifying quadrilaterals

Which of the following are parallelograms?

A B
C D
Answer: The parallelograms are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a parallelograms? They could have extra properties as well - that does not mean they are not parallelograms.


STEP: Identify the quadrilaterals that meet the definition of a parallelogram
[−2 points ⇒ 0 / 2 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.

We can easily recognise Quadrilateral B as a parallelogram. However, this is not the only type of parallelogram in the options.

Quadrilaterals C and D each have two pairs of opposite sides that are parallel. So, C and D are also types of parallelograms. This is true even though the exact name for C is a rectangle, and the exact name for D is a square (because they have a few extra properties too).


Submit your answer as:

ID is: 4216 Seed is: 7590

Identifying quadrilaterals

Which of the following are parallelograms?

A B
C D
Answer: The parallelograms are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a parallelograms? They could have extra properties as well - that does not mean they are not parallelograms.


STEP: Identify the quadrilaterals that meet the definition of a parallelogram
[−2 points ⇒ 0 / 2 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.

We can easily recognise Quadrilateral A as a parallelogram. However, this is not the only type of parallelogram in the options.

Quadrilaterals B and C each have two pairs of opposite sides that are parallel. So, B and C are also types of parallelograms. This is true even though the exact name for B is a rhombus, and the exact name for C is a square (because they have a few extra properties too).


Submit your answer as:

ID is: 1846 Seed is: 4706

Angles on a straight line

Line BC represents angles on one side of a straight line. a = 58° , b = x and c = 58°.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°58°+x+58°=180°x=64°

Submit your answer as: and

ID is: 1846 Seed is: 6883

Angles on a straight line

Line YZ represents angles on one side of a straight line. a = 46° , b = 54° and c = x.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°46°+54°+x=180°x=80°

Submit your answer as: and

ID is: 1846 Seed is: 8263

Angles on a straight line

Line BC represents angles on one side of a straight line. a = 68° , b = 70° and c = x.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°68°+70°+x=180°x=42°

Submit your answer as: and

ID is: 1864 Seed is: 2508

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because BC is a straight line. We can write an equation based on this information:

x+108°+40°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

x+108°+40°=180°x=32°

The complete diagram, with all three angles known, is:

The missing angle is 32°, which is acute.


Submit your answer as: and

ID is: 1864 Seed is: 2102

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because YZ is a straight line. We can write an equation based on this information:

50°+x+50°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

50°+x+50°=180°x=80°

The complete diagram, with all three angles known, is:

The missing angle is 80°, which is acute.


Submit your answer as: and

ID is: 1864 Seed is: 6528

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because ST is a straight line. We can write an equation based on this information:

57°+x+60°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

57°+x+60°=180°x=63°

The complete diagram, with all three angles known, is:

The missing angle is 63°, which is acute.


Submit your answer as: and

ID is: 1672 Seed is: 3157

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral XWVU, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles X,W,V and U). So X1=X2, W1=W2, and so on. These dashed lines form quadrilateral EFGH inside of parallelogram XWVU.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial EFGH is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
X=V Opposite angles in a parallelogram are equal
W=U Opposite angles in a parallelogram are equal
W2=U1 and W1=U2 Angles created by bisectors of equal angles are equal
XU=WV Opposite sides of a parallelogram are equal
In quadrilateral EFGH:E2=G2 Congruent triangles have equal angles (ΔXUEΔVGW)
H1=F1 Congruent triangles have equal angles (ΔXHWΔVFU)
H1=H2 and F1=F2 Vertically opposite angles are equal
EFGH is a parallelogram Opposite angles in a parallelogram are equal
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, EFGH. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral XWVU is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that EFGH is a parallelogram. We can do that if we can show that the opposite angles inside EFGH are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that EFGH has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral EFGH is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
X=V Opposite angles in a parallelogram are equal
X2=V1 and X1=V2 Angles created by bisectors of equal angles are equal
W=U Opposite angles in a parallelogram are equal
W2=U1 and W1=U2 Angles created by bisectors of equal angles are equal
XU=WV Opposite sides of a parallelogram are equal
ΔXUEΔVGW Congruent triangles (Angle-Side-Angle)
In quadrilateral EFGH:E2=G2 Congruent triangles have equal angles (ΔXUEΔVGW)
XW=UV Opposite sides of a parallelogram are equal
ΔXHWΔVFU Congruent triangles (Angle-Side-Angle)
H1=F1 Congruent triangles have equal angles (ΔXHWΔVFU)
H1=H2 and F1=F2 Vertically opposite angles are equal
EFGH is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

ID is: 1672 Seed is: 7478

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral ABCD, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles A,B,C and D). So A1=A2, B1=B2, and so on. These dashed lines form quadrilateral MNOP inside of parallelogram ABCD.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial MNOP is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
A=C
A2=C1 and A1=C2
B=D Opposite angles in a parallelogram are equal
Angles created by bisectors of equal angles are equal
AD=BC Opposite sides of a parallelogram are equal
Congruent triangles (Angle-Side-Angle)
Congruent triangles have equal angles (ΔADMΔCOB)
AB=DC Opposite sides of a parallelogram are equal
ΔAPBΔCND
Congruent triangles have equal angles (ΔAPBΔCND)
P1=P2 and N1=N2
MNOP is a parallelogram Opposite angles in a parallelogram are equal
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, MNOP. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral ABCD is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that MNOP is a parallelogram. We can do that if we can show that the opposite angles inside MNOP are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that MNOP has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral MNOP is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
A=C Opposite angles in a parallelogram are equal
A2=C1 and A1=C2 Angles created by bisectors of equal angles are equal
B=D Opposite angles in a parallelogram are equal
B2=D1 and B1=D2 Angles created by bisectors of equal angles are equal
AD=BC Opposite sides of a parallelogram are equal
ΔADMΔCOB Congruent triangles (Angle-Side-Angle)
In quadrilateral MNOP:M2=O2 Congruent triangles have equal angles (ΔADMΔCOB)
AB=DC Opposite sides of a parallelogram are equal
ΔAPBΔCND Congruent triangles (Angle-Side-Angle)
P1=N1 Congruent triangles have equal angles (ΔAPBΔCND)
P1=P2 and N1=N2 Vertically opposite angles are equal
MNOP is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

ID is: 1672 Seed is: 8308

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral QRST, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles Q,R,S and T). So Q1=Q2, R1=R2, and so on. These dashed lines form quadrilateral JKLM inside of parallelogram QRST.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial JKLM is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
Q2=S1 and Q1=S2 Angles created by bisectors of equal angles are equal
R=T Opposite angles in a parallelogram are equal
Angles created by bisectors of equal angles are equal
QT=RS Opposite sides of a parallelogram are equal
In quadrilateral JKLM:J2=L2 Congruent triangles have equal angles (ΔQTJΔSLR)
QR=TS Opposite sides of a parallelogram are equal
ΔQMRΔSKT Congruent triangles (Angle-Side-Angle)
Congruent triangles have equal angles (ΔQMRΔSKT)
M1=M2 and K1=K2
JKLM is a parallelogram
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, JKLM. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral QRST is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that JKLM is a parallelogram. We can do that if we can show that the opposite angles inside JKLM are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that JKLM has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral JKLM is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
Q=S Opposite angles in a parallelogram are equal
Q2=S1 and Q1=S2 Angles created by bisectors of equal angles are equal
R=T Opposite angles in a parallelogram are equal
R2=T1 and R1=T2 Angles created by bisectors of equal angles are equal
QT=RS Opposite sides of a parallelogram are equal
ΔQTJΔSLR Congruent triangles (Angle-Side-Angle)
In quadrilateral JKLM:J2=L2 Congruent triangles have equal angles (ΔQTJΔSLR)
QR=TS Opposite sides of a parallelogram are equal
ΔQMRΔSKT Congruent triangles (Angle-Side-Angle)
M1=K1 Congruent triangles have equal angles (ΔQMRΔSKT)
M1=M2 and K1=K2 Vertically opposite angles are equal
JKLM is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

ID is: 1866 Seed is: 4127

Angles in a full turn

In the diagram below, line YZ is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line YZ is a straight line, and angles on a straight line up to 180°. This means that all of the angles above YZ add up to 180°:

10°+x+98°=180°(s on a str line)x=72°

In the same way, all of the angles below YZ add up to 180°:

34°+58°+y+51°=180°(s on a str line)y=37°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

10°+72°+98°+34°+58°+37°+51°=360°

y=37°

y is an acute angle (less than 90°).


Submit your answer as: andand

ID is: 1866 Seed is: 2125

Angles in a full turn

In the diagram below, line YZ is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line YZ is a straight line, and angles on a straight line up to 180°. This means that all of the angles above YZ add up to 180°:

35°+82°+x=180°(s on a str line)x=63°

In the same way, all of the angles below YZ add up to 180°:

46°+36°+23°+y=180°(s on a str line)y=75°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

35°+82°+63°+46°+36°+23°+75°=360°

y=75°

y is an acute angle (less than 90°).


Submit your answer as: andand

ID is: 1866 Seed is: 7078

Angles in a full turn

In the diagram below, line BC is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line BC is a straight line, and angles on a straight line up to 180°. This means that all of the angles above BC add up to 180°:

40°+x+13°=180°(s on a str line)x=127°

In the same way, all of the angles below BC add up to 180°:

95°+44°+18°+y=180°(s on a str line)y=23°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

40°+127°+13°+95°+44°+18°+23°=360°

y=23°

y is an acute angle (less than 90°).


Submit your answer as: andand

ID is: 4181 Seed is: 2204

Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a square.

Answer: A square is a rhombus with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a rhombus? What additional properties does a square have?


STEP: Compare the definitions for a square and a rhombus
[−1 point ⇒ 0 / 1 points left]
A rhombus is a quadrilateral with adjacent sides equal and two pairs of opposite sides parallel.
A square is a quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90°.

The following diagram shows a rhombus next to a square, with their defining properties filled in.

We can see that the square has the same properties as the rhombus, plus corner angles 90°.


Submit your answer as:

ID is: 4181 Seed is: 4747

Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a square.

Answer: A square is a rhombus with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a rhombus? What additional properties does a square have?


STEP: Compare the definitions for a square and a rhombus
[−1 point ⇒ 0 / 1 points left]
A rhombus is a quadrilateral with two pairs of opposite sides parallel and adjacent sides equal.
A square is a quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90°.

The following diagram shows a rhombus next to a square, with their defining properties filled in.

We can see that the square has the same properties as the rhombus, plus corner angles 90°.


Submit your answer as:

ID is: 4181 Seed is: 8441

Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a rectangle.

Answer: A rectangle is a parallelogram with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a parallelogram? What additional properties does a rectangle have?


STEP: Compare the definitions for a rectangle and a parallelogram
[−1 point ⇒ 0 / 1 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides parallel.
A rectangle is a quadrilateral with two pairs of opposite sides parallel and all corners 90°.

The following diagram shows a parallelogram next to a rectangle, with their defining properties filled in.

NOTE: It is true that the opposite sides of a rectangle are the same length. But, this is also true for all parallelograms. So, we do not say that a rectangle is a parallelogram with opposite sides equal.

We can see that the rectangle has the same properties as the parallelogram, plus corner angles 90°.


Submit your answer as:

ID is: 4300 Seed is: 8349

Using diagonals to prove properties

In the diagram below, WVXY is a quadrilateral with WV=VX and WVZ=XVZ=39°.

  1. How should we prove that WY^Z=XY^Z?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    WY^Z is a angle in ΔWVY and XY^Z is the matching angle in ΔXVY. So, if we prove that ΔWVYΔXVY , then WY^Z and XY^Z must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔWVYΔXVY . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that WY^Z=XY^Z by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. WVZ=XVZ (given)

    WY^Z=XY^Z

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔWVY and ΔXVY:

    1. VY is common.
    2. WV=VX (given)
    3. WVZ=XVZ (given)

    ΔWVYΔXVY (SAS)

    WY^Z=XY^Z (ΔWVYΔXVY)

    NOTE: You can extend this proof to prove that the diagonals of any rhombus bisect all of its vertices.

    Submit your answer as: andandandandand

ID is: 4300 Seed is: 417

Using diagonals to prove properties

In the diagram below, WVXY is a quadrilateral with WV=VX and WVZ=XVZ=24°.

  1. How should we prove that WY^Z=XY^Z?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    WY^Z is a angle in ΔWVY and XY^Z is the matching angle in ΔXVY. So, if we prove that ΔWVYΔXVY , then WY^Z and XY^Z must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔWVYΔXVY . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that WY^Z=XY^Z by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. WVZ=XVZ (given)

    WY^Z=XY^Z

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔWVY and ΔXVY:

    1. VY is common.
    2. WV=VX (given)
    3. WVZ=XVZ (given)

    ΔWVYΔXVY (SAS)

    WY^Z=XY^Z (ΔWVYΔXVY)

    NOTE: You can extend this proof to prove that the diagonals of any rhombus bisect all of its vertices.

    Submit your answer as: andandandandand

ID is: 4300 Seed is: 8103

Using diagonals to prove properties

In the diagram below, WVXY is a quadrilateral with WV=VX and WVZ=XVZ=29°.

  1. How should we prove that WY^Z=XY^Z?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    WY^Z is a angle in ΔWVY and XY^Z is the matching angle in ΔXVY. So, if we prove that ΔWVYΔXVY , then WY^Z and XY^Z must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔWVYΔXVY . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that WY^Z=XY^Z by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. WVZ=XVZ (given)

    WY^Z=XY^Z

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔWVY and ΔXVY:

    1. VY is common.
    2. WV=VX (given)
    3. WVZ=XVZ (given)

    ΔWVYΔXVY (SAS)

    WY^Z=XY^Z (ΔWVYΔXVY)

    NOTE: You can extend this proof to prove that the diagonals of any rhombus bisect all of its vertices.

    Submit your answer as: andandandandand

ID is: 1660 Seed is: 5416

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral QRST with Q=S=59° and R=T=121°.

The steps and reasons below prove that QRST is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
Given (59°+121°=180°)
QRTS
T+S=180°
Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram Definition of a parallelogram
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side QR is parallel to side TS and that side RS is parallel to side QT.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral QRST is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
Q+T=180° Given (59°+121°=180°)
QRTS Co-interior angles of parallel lines have a sum of 180°
T+S=180° Given (121°+59°=180°)
RSQT Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

ID is: 1660 Seed is: 2940

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral XWVU with X=V=74° and W=U=106°.

The steps and reasons below prove that XWVU is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
Given (74°+106°=180°)
Co-interior angles of parallel lines have a sum of 180°
U+V=180°
WVXU
XWVU is a parallelogram Definition of a parallelogram
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side XW is parallel to side UV and that side WV is parallel to side XU.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral XWVU is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
X+U=180° Given (74°+106°=180°)
XWUV Co-interior angles of parallel lines have a sum of 180°
U+V=180° Given (106°+74°=180°)
WVXU Co-interior angles of parallel lines have a sum of 180°
XWVU is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

ID is: 1660 Seed is: 8838

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral ABCD with A=C=121° and B=D=59°.

The steps and reasons below prove that ABCD is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
Given (121°+59°=180°)
ABDC Co-interior angles of parallel lines have a sum of 180°
D+C=180°
Co-interior angles of parallel lines have a sum of 180°
ABCD is a parallelogram
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side AB is parallel to side DC and that side BC is parallel to side AD.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral ABCD is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
A+D=180° Given (121°+59°=180°)
ABDC Co-interior angles of parallel lines have a sum of 180°
D+C=180° Given (59°+121°=180°)
BCAD Co-interior angles of parallel lines have a sum of 180°
ABCD is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

ID is: 4215 Seed is: 4463

Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

All kites are squares.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a kite always have at least the same properties as a square? Or, can you think of any kites that are not types of squares?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]
A kite is a quadrilateral with two pairs of equal adjacent sides.
A square is a quadrilateral with two pairs of opposite sides that are parallel, equal adjacent sides, and corners that are 90°.

Consider the following kite:

This shape meets the definition of a kite. But, it does not meet the definition of a square because the opposite sides are not parallel (amongst other things). So, we have come up with an example that disagrees with the statement. Therefore, the statement is false.

NOTE: An example that disagrees with the statement is called a counter-example. We only need one counter-example to prove that a statement is false. This is because if it is false for one example, then we cannot say that it is true for all - in other words, it is false.

Submit your answer as:

ID is: 4215 Seed is: 37

Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

All squares are rhombuses.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a square always have at least the same properties as a rhombus? Or, can you think of any squares that are not types of rhombuses?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]
A square is a quadrilateral with two pairs of opposite sides that are parallel, equal adjacent sides, and corners that are 90°.
A rhombus is a quadrilateral with two pairs of opposite sides that are parallel and equal adjacent sides.

We can see that a square has all of the same properties as a rhombus (and some more). So, squares are special types of rhombuses. Therefore, the statement is true.

NOTE: It does not matter that a square has extra properties that a rhombus does not have. What matters is that it has at least the properties of a rhombus.

Submit your answer as:

ID is: 4215 Seed is: 141

Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

All squares are kites.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a square always have at least the same properties as a kite? Or, can you think of any squares that are not types of kites?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]
A square is a quadrilateral with two pairs of opposite sides that are parallel, equal adjacent sides, and corners that are 90°.
A kite is a quadrilateral with two pairs of equal adjacent sides.

We can see that a square has all of the same properties as a kite (and some more). So, squares are special types of kites. Therefore, the statement is true.

NOTE: It does not matter that a square has extra properties that a kite does not have. What matters is that it has at least the properties of a kite.

Submit your answer as:

ID is: 4299 Seed is: 7598

Proving diagonals using congruency

In the diagram below, WVXY is a quadrilateral with WV=XY and WVXY .

Prove that ZW=ZX.

  1. How should we prove that ZW=ZX?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔYZXΔVZW.

    ZW is a side in ΔVZW. ZX is the matching side in ΔYZX. So, if we prove that ΔYZXΔVZX, then ZW and ZX must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔYZXΔVZX. See if you can spot one pair of sides and two pairs of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that ZW=ZX by selecting the correct options.

    Answer:

    In :

    1. WV=XY (given)
    2. ZVW=
    3. ZWV=

    (SAA)
    ZW=ZX

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔYZX and ΔVZW:

    1. WV=XY (given)
    2. ZVW=ZYX (alt s; WVXY)
    3. ZWV=ZXY (alt s; WVXY)

    ΔYZXΔVZW (SAA)

    ZW=ZX (ΔYZXΔVZW)

    NOTE: You can extend this proof to prove that diagonals bisect each other in all parallelograms. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandandand

ID is: 4299 Seed is: 9718

Proving diagonals using congruency

In the diagram below, ABCD is a quadrilateral with AB=CD and ABCD .

Prove that EA=EC.

  1. How should we prove that EA=EC?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔDECΔBEA.

    EA is a side in ΔBEA. EC is the matching side in ΔDEC. So, if we prove that ΔDECΔBEC, then EA and EC must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔDECΔBEC. See if you can spot one pair of sides and two pairs of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral ABCD is repeated for convenience:

    Now, prove that EA=EC by selecting the correct options.

    Answer:

    In :

    1. AB=CD (given)
    2. EBA=
    3. EAB=

    (SAA)
    EA=EC

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔDEC and ΔBEA:

    1. AB=CD (given)
    2. EBA=EDC (alt s; ABCD)
    3. EAB=ECD (alt s; ABCD)

    ΔDECΔBEA (SAA)

    EA=EC (ΔDECΔBEA)

    NOTE: You can extend this proof to prove that diagonals bisect each other in all parallelograms. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandandand

ID is: 4299 Seed is: 8785

Proving diagonals using congruency

In the diagram below, PQRS is a quadrilateral with PS=QR and PQ^R=QP^S= 90°.

Prove that PR=QS.

  1. How should we prove that PR=QS?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔPQSΔQPR.

    PR is a side in ΔQPR and QS is the matching side in ΔPQS. So, if we prove that ΔPQSΔQPR, then PR and QS must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔPQSΔQPR. See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral PQRS is repeated for convenience:

    Now, prove that PR=QS by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. PS=QR
    3. PQR=SPQ=90° (given)


    PR=QS

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔPQS and ΔQPR :

    1. PQ is common.
    2. PS=QR (given)
    3. PQR=SPQ=90° (given)

    ΔPQSΔQPR (SAS)

    PR=QS (ΔPQSΔQPR)

    NOTE: You can extend this proof to prove that diagonals are equal in all rectangles. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandand

ID is: 1862 Seed is: 1427

Calculate angles between parallel lines

In the diagram below, FG HJ. KL is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Corresponding angles are in the same position compared to the parallel lines and the transversal. If we highlight the angles we will see an "F" shape. This helps us to identify corresponding angles.

Since we know that corresponding angles are equal, we can write:

x=104°(corresp s,FGHJ)

Therefore x=104° (corresp s, FGHJ).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

ID is: 1862 Seed is: 6355

Calculate angles between parallel lines

In the diagram below, CD EF. GH is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Alternate angles are both inside the parallel lines, but on opposite sides of the transversal. If we highlight the angles we will see a "Z" shape. This helps us to identify alternate angles.

Since we know that alternate angles are equal, we can write:

x=50°(alt s,CDEF)

Therefore x=50° (alt s, CDEF).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

ID is: 1862 Seed is: 5441

Calculate angles between parallel lines

In the diagram below, DE FG. HJ is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Alternate angles are both inside the parallel lines, but on opposite sides of the transversal. If we highlight the angles we will see a "Z" shape. This helps us to identify alternate angles.

Since we know that alternate angles are equal, we can write:

x=96°(alt s,DEFG)

Therefore x=96° (alt s, DEFG).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

ID is: 4252 Seed is: 7898

Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral:

The diagonals are perpendicular to each other.
Answer:

The statement is definitely true for all:

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
[−2 points ⇒ 0 / 2 points left]

Perpendicular means that the diagonals meet at 90°. This is true for all kites.

NOTE: This is true for all kites, so it must also be true for rhombuses and squares (which are types of kites).

Submit your answer as:

ID is: 4252 Seed is: 1311

Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral:

The diagonals are perpendicular bisectors of each other.
Answer:

The statement is definitely true for all:

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
[−2 points ⇒ 0 / 2 points left]

Perpendicular means that the diagonals meet at 90°. This is true for all kites.

The diagonals of all parallelograms bisect each other.

NOTE: Bisect means to cut into two equal parts.

So, quadrilaterals that have these properties must be types of kites, and also types of parallelograms. So, the quadrilaterals for which this is definitely true are rhombuses.

NOTE: This is true for all rhombuses, so it must also be true for squares (which are types of rhombuses).

Submit your answer as:

ID is: 4252 Seed is: 6937

Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral:

The diagonals are equal in length.
Answer:

The statement is definitely true for all:

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
[−2 points ⇒ 0 / 2 points left]

The diagonals of all rectangles are equal in length. So, the quadrilateral must be a rectangle.

NOTE: This is true for all rectangles, so it must also be true for squares (which are types of rectangles).

Submit your answer as:

ID is: 1867 Seed is: 1142

Recognise pairs of angles between parallel lines

In the diagram below, ABCD.  EF is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates corresponding angles.

Corresponding angles are in the same position compared to the parallel lines and the transversal. If we highlight the angles we will see an "F" shape. This helps us to identify corresponding angles.


Submit your answer as:

ID is: 1867 Seed is: 5144

Recognise pairs of angles between parallel lines

In the diagram below, ABCD.  EF is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates co-interior angles.

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us identify co-interior angles.


Submit your answer as:

ID is: 1867 Seed is: 7432

Recognise pairs of angles between parallel lines

In the diagram below, ABCD.  EF is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates co-interior angles.

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us identify co-interior angles.


Submit your answer as:

ID is: 4276 Seed is: 6678

Congruency in kites

  1. Consider the following diagram. ABCD is a kite.

    Prove that ΔABEΔADE by completing the proof below.

    Answer:

    In ΔABE and ΔADE:

    1. AB=
    2. =DE
    3. is common

    ΔABEΔADE

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔABE and ΔADE:

    1. AB=AD (adjacent sides of kite)
    2. BE=DE (given)
    3. AE is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔABEΔADE (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram. ABCD is a kite.

    You have just proved that ΔABEΔADE. Hence, determine the size of AD^E.

    Answer: AD^E= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches AD^E?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔABEΔADE.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, AD^E=AB^E.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    AD^E= 40° (ΔABEΔADE)


    Submit your answer as: and

ID is: 4276 Seed is: 6046

Congruency in kites

  1. Consider the following diagram. PQRS is a kite.

    Prove that ΔPQRΔPSR by completing the proof below.

    Answer:

    In ΔPQR and ΔPSR:

    1. PQ=
    2. =SR
    3. is common

    ΔPQRΔPSR

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔPQR and ΔPSR:

    1. PQ=PS (adjacent sides of kite)
    2. QR=SR (adjacent sides of kite)
    3. PR is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔPQRΔPSR (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram. PQRS is a kite.

    You have just proved that ΔPQRΔPSR. Hence, determine the size of PR^S.

    Answer: PR^S= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches PR^S?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔPQRΔPSR.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, PR^S=QR^P.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    PR^S= 23° (ΔPQRΔPSR)


    Submit your answer as: and

ID is: 4276 Seed is: 4410

Congruency in kites

  1. Consider the following diagram. CBAD is a kite.

    Prove that ΔCBEΔCDE by completing the proof below.

    Answer:

    In ΔCBE and ΔCDE:

    1. CB=
    2. =DE
    3. is common

    ΔCBEΔCDE

    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔCBE and ΔCDE:

    1. CB=CD (adjacent sides of kite)
    2. BE=DE (given)
    3. CE is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔCBEΔCDE (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram. CBAD is a kite.

    You have just proved that ΔCBEΔCDE. Hence, determine the size of CD^E.

    Answer: CD^E= °
    numeric
    2 attempts remaining
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches CD^E?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔCBEΔCDE.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, CD^E=CB^E.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    CD^E= 42° (ΔCBEΔCDE)


    Submit your answer as: and

4. Solving riders


ID is: 4154 Seed is: 2953

Algebraic triangles

Determine the size of y, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for R^ in terms of y.
Answer: R^=
y= °
expression
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found R^ in terms of y, write an equation for this triangle.


STEP: Find a missing angle in terms of y
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

R^=y(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know all the angles of the triangle in terms of y, so we can write an equation using the sum of angles inside the triangle.

y+y+3y50°=180°(sum of s in Δ)

STEP: Solve the equation to determine the value of y
[−1 point ⇒ 0 / 5 points left]
5y50°=180°5y50°+50°=180°+50°5y=230°5y5=230°5y=46°

Submit your answer as: andandand

ID is: 4154 Seed is: 1322

Algebraic triangles

Determine the size of y, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for C^ in terms of y.
Answer: C^=
y= °
expression
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found C^ in terms of y, write an equation for this triangle.


STEP: Find a missing angle in terms of y
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

C^=2y(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know all the angles of the triangle in terms of y, so we can write an equation using the sum of angles inside the triangle.

2y+2y+2y+12°=180°(sum of s in Δ)

STEP: Solve the equation to determine the value of y
[−1 point ⇒ 0 / 5 points left]
6y+12°=180°6y+12°12°=180°12°6y=168°6y6=168°6y=28°

Submit your answer as: andandand

ID is: 4154 Seed is: 8298

Algebraic triangles

Determine the size of z, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for C^ in terms of z.
Answer: C^=
z= °
expression
numeric
2 attempts remaining
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found C^ in terms of z, write an equation for this triangle.


STEP: Find a missing angle in terms of z
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

C^=3z(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know all the angles of the triangle in terms of z, so we can write an equation using the sum of angles inside the triangle.

3z+3z+5z18°=180°(sum of s in Δ)

STEP: Solve the equation to determine the value of z
[−1 point ⇒ 0 / 5 points left]
11z18°=180°11z18°+18°=180°+18°11z=198°11z11=198°11z=18°

Submit your answer as: andandand